Ethylene reacts with benzene producing ethylbenzene according to the following reaction:

$C_2{H}_2+C_6{H}_6{C}_6{H}_5{C}_2{H}_5,(E+BEB)$

The fresh feed contains $1000$ moles of ethylene, $1000$ moles of benzene and $300$ moles of inerts. The fresh feed is mixed with a recycle stream to produce the feed to the reactor. The conversion of ethylene in the reactor is $70\%.$ The effluent from the reactor is fed to a separator to separate product EB from unreacted E and B$.$ The separator is designed so that the top product contains $90\%$ of ethylene, $80\%$ of benzene $5\%$ of ethylbenzene and $70\%$ of the inerts in the feed to the separator. The top product from the separator is recycled and mixed with the fresh feed.

Perform degrees of freedom analysis and calculate all the missing flow rates and compositions for the following scenarios:

a$.$ Recycling all the separator top product.

b$.$ Purging $30\%$ of the separator top product.

c$.$ Resolve part a given that the conversion of ethylene in the reactor is $90\%($instead of $70\%).$

After solving the three scenarios, compare the inerts concentration in the reactor feed stream between cases a and $b,$ and compare the flow rate of the recycle stream in cases a and $c.$